Potential Polynomials and Motzkin Paths

TL;DR

This paper explores the combinatorial properties of Motzkin paths, focusing on the statistics of u- and h-segments, and employs generating functions and potential polynomials to analyze their distributions.

Contribution

It introduces a novel framework using potential polynomials and Lagrange inversion to analyze segment statistics in Motzkin paths, extending to compositions.

Findings

Derived weighted generating functions for Motzkin path statistics

Expressed generating functions as sums of Bell and potential polynomials

Provided a general framework for studying compositions

Abstract

A {\em Motzkin path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of horizontal-steps $(1, 0)$, up-steps $(1,1)$, and down-steps $(1,-1)$, which never passes below the x-axis. A {\em $u$-segment {\rm (resp.} $h$-segment {\rm)}} of a Motzkin path is a maximum sequence of consecutive up-steps ({\rm resp.} horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: "number of $u$-segments" and "number of $h$-segments". The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. As an application, a general framework for studying compositions are also provided.